Easiest way to kinda botch your way through this is sticking them on a larger size media and making that go through. If you can, get the same card stock and make a cutout to fit it in to have the whole sheet the same thickness.

I do not recommend any of this but it's the only solution on the top of my head without getting a new printer or batch of cards.

Other solution is that business cards are really cheap now online... Might be worth it to scour for a deal and go with that instead..

I suspect you may want to look into outsourcing this kind of printing

You need to define what the fliers look like. Are they full page color? Or mostly white? Must they be borderless?

Must the prints stand up to getting wet? Or is it fine if the ink runs

Are there photos on the flier? Does the quality of the photo matter, or is it ok if the colors on the photo are somewhat wrong

How much do you want to pay per flier?

I work for a commercial printer (as a web developer). What you want requires a reasonable quality digital printer, a cutter, a corner rounder, a die cutter, a way to make dies, a scoring machine (though the die cutter can probably score), some form of shrink wrapper, and the ability to ship at a reasonable price. Those machines all run on 240 volt and up. Our shop has a 640 volt main, so I'm guessing some machines use that. You wont be able to do that at home. Also, new, you are looking at probably $1 million in equipment alone. Used, can be significantly cheaper, but some machines (cutter, die cutter, corner rounder) will be old, made of solid iron and weigh literally tons. We sold an old cutter and had to hire a specialty engineering firm to work out how to move it.

So, your best bet is to find a commercial printer. You are in luck because the current trend is a downward pressure on price (aka. Prices are low). However, finding a commercial printer can be difficult because many of them don't advertise. I would visit your local industrial district and drive around looking for print shops, or call local companies and ask who their print vendor is. Avoid the corner copy shop, sir speedy or kinkos. Hight prices and low quality.